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Ordinary Interactive SmallStep Algorithms
 I,” ACM Trans. Computational Logic
, 2004
"... This is the first in a series of papers extending the Abstract State Machine Thesis — that arbitrary algorithms are behaviorally equivalent to abstract state machines — to algorithms that can interact with their environments during a step rather than only between steps. In the present paper, we desc ..."
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Cited by 31 (16 self)
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This is the first in a series of papers extending the Abstract State Machine Thesis — that arbitrary algorithms are behaviorally equivalent to abstract state machines — to algorithms that can interact with their environments during a step rather than only between steps. In the present paper, we describe, by means of suitable postulates, those interactive algorithms that (1) proceed in discrete, global steps, (2) perform only a bounded amount of work in each step, (3) use only such information from the environment as can be regarded as answers to queries, and (4) never complete a step until all queries from that step have been answered. We indicate how a great many sorts of interaction meet these requirements. We also discuss in detail the structure of queries and replies and the appropriate definition of equivalence of algorithms. Finally, motivated by our considerations concerning queries, we discuss a generalization of firstorder logic in which the arguments of function and relation symbols are not merely tuples of elements but orbits of such tuples under groups of permutations of the argument places.
Algorithms: A quest for absolute definitions
 Bulletin of the European Association for Theoretical Computer Science
, 2003
"... y Abstract What is an algorithm? The interest in this foundational problem is not only theoretical; applications include specification, validation and verification of software and hardware systems. We describe the quest to understand and define the notion of algorithm. We start with the ChurchTurin ..."
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Cited by 19 (9 self)
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y Abstract What is an algorithm? The interest in this foundational problem is not only theoretical; applications include specification, validation and verification of software and hardware systems. We describe the quest to understand and define the notion of algorithm. We start with the ChurchTuring thesis and contrast Church's and Turing's approaches, and we finish with some recent investigations.
Abstract State Machines: A unifying view of models of computation and of system design frameworks
 Annals of Pure and Applied Logic
, 2005
"... We capture the principal models of computation and specification in the literature by a uniform set of transparent mathematical descriptions which—starting from scratch—provide the conceptual basis for a comparative study 1. 1 ..."
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Cited by 9 (5 self)
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We capture the principal models of computation and specification in the literature by a uniform set of transparent mathematical descriptions which—starting from scratch—provide the conceptual basis for a comparative study 1. 1
The hidden computation steps of turbo Abstract State Machines
 Abstract State Machines — Advances in Theory and Applications, 10th International Workshop, ASM 2003
, 2003
"... Abstract. Turbo Abstract State Machines are ASMs with parallel and sequential composition and possibly recursive submachine calls. Turbo ASMs are viewed as blackboxes that can combine arbitrary many steps of one or more submachines into one big step. The intermediate steps of a turbo ASM are not ob ..."
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Cited by 8 (2 self)
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Abstract. Turbo Abstract State Machines are ASMs with parallel and sequential composition and possibly recursive submachine calls. Turbo ASMs are viewed as blackboxes that can combine arbitrary many steps of one or more submachines into one big step. The intermediate steps of a turbo ASM are not observable from outside. It is not even clear what exactly the intermediate steps are, because the semantics of turbo ASMs is usually defined inductively along the call graph of the ASM and the structure of the rule bodies. The most important application of turbo ASMs are recursive algorithms. Such algorithms can directly be simulated on turbo ASMs without transforming them into multiagent (distributed) ASMs. In this article we analyze the hidden intermediate steps of turbo ASMs and characterize them using PAR/SEQ trees. We also address the problem of the reserve in the presence of recursion and sequential composition. 1
Polygraphic programs and polynomialtime functions
"... Abstract – We study the computational model of polygraphs. For that, we consider polygraphic programs, a subclass of these objects, as a formal description of firstorder functional programs. We explain their semantics and prove that they form a Turingcomplete computational model. Their algebraic s ..."
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Cited by 5 (0 self)
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Abstract – We study the computational model of polygraphs. For that, we consider polygraphic programs, a subclass of these objects, as a formal description of firstorder functional programs. We explain their semantics and prove that they form a Turingcomplete computational model. Their algebraic structure is used by analysis tools, called polygraphic interpretations, for complexity analysis. In particular, we delineate a subclass of polygraphic programs that compute exactly the functions that are Turingcomputable in polynomial time.
On primitive recursive algorithms and the greatest common divisor function
 Theor. Comput. Sci
, 2003
"... Abstract. We establish linear lower bounds for the complexity of nontrivial, primitive recursive algorithms from piecewise linear given functions. The main corollary is that logtime algorithms for the greatest common divisor from such givens (such as Stein’s) cannot be matched in efficiency by prim ..."
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Cited by 4 (2 self)
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Abstract. We establish linear lower bounds for the complexity of nontrivial, primitive recursive algorithms from piecewise linear given functions. The main corollary is that logtime algorithms for the greatest common divisor from such givens (such as Stein’s) cannot be matched in efficiency by primitive recursive algorithms from the same given functions. The question is left open for the Euclidean algorithm, which assumes the remainder function. In 1991, Colson [3] 1 proved a remarkable theorem about the limitations of primitive recursive algorithms, which has the following consequence: Colson’s Corollary. If a primitive recursive derivation of min(x, y) is expressed faithfully in a programming language, then one of the two computations min(1, 1000) and min(1000, 1) will take at least 1000 steps. The point is that the natural algorithm which computes min(x, y) in O(min(x, y)) steps cannot be matched in efficiency by a primitive recursive program, even though min(x, y) is a primitive recursive function; and so, as a practical and (especially) a foundational matter, we need to consider “recursive schemes ” more general than primitive recursion, even if, ultimately, we are only interested in primitive recursive functions. In this paper we consider extensions of Colson’s Theorem which allow conditional definitions and especially calls to a rich variety of “given ” functions, whose values are produced on demand in constant time. Sample, easy to state, result: Corollary 20. Consider primitiverecursivelike derivations, which in addition to composition and primitive recursion allow definition by cases and calls to the following functions and (characteristic functions of) relations: x + y, x − · y, x ÷ 2, Parity(x), x = y, x < y For each such derivation of the greatest common divisor function gcd(x, y), there is a sequence of pairs {(xt, yt)} and a rational constant r> 0, such that limt(xt + yt) = ∞, and for all t, c ∗ (xt, yt) ≥ r(xt + yt),
Interactive smallstep algorithms I: Axiomatization,
, 2006
"... In earlier work, the Abstract State Machine Thesis — that arbitrary algorithms are behaviorally equivalent to abstract state machines — was established for several classes of algorithms, including ordinary, interactive, smallstep algorithms. This was accomplished on the basis of axiomatizations o ..."
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Cited by 4 (2 self)
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In earlier work, the Abstract State Machine Thesis — that arbitrary algorithms are behaviorally equivalent to abstract state machines — was established for several classes of algorithms, including ordinary, interactive, smallstep algorithms. This was accomplished on the basis of axiomatizations of these classes of algorithms. Here we extend the axiomatization and, in a companion paper, the proof, to cover interactive smallstep algorithms that are not necessarily ordinary. This means that the algorithms (1) can complete a step without necessarily waiting for replies to all queries from that step and (2) can use not only the environment’s replies but also the order in which the replies were received.
Computation and specification models. A comparative study
 Department of Computer Science at University of Aarhus
, 2002
"... For each of the principal current models of computation and of highlevel system design, we present a uniform set of transparent easily understandable descriptions, which are faithful to the basic intuitions and concepts of the investigated systems. Our main goal is to provide a mathematical basis fo ..."
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Cited by 4 (2 self)
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For each of the principal current models of computation and of highlevel system design, we present a uniform set of transparent easily understandable descriptions, which are faithful to the basic intuitions and concepts of the investigated systems. Our main goal is to provide a mathematical basis for the technical comparison of established models of computation which can contribute to rationalize the scientific evaluation of different system specification approaches in the literature, clarifying in detail their advantages and disadvantages. As a side effect we obtain a powerful yet simple new conceptual framework for teaching the fundamentals of computation theory. 1
A natural axiomatization of Church’s thesis
, 2007
"... The Abstract State Machine Thesis asserts that every classical algorithm is behaviorally equivalent to an abstract state machine. This thesis has been shown to follow from three natural postulates about algorithmic computation. Here, we prove that augmenting those postulates with an additional requ ..."
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Cited by 2 (0 self)
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The Abstract State Machine Thesis asserts that every classical algorithm is behaviorally equivalent to an abstract state machine. This thesis has been shown to follow from three natural postulates about algorithmic computation. Here, we prove that augmenting those postulates with an additional requirement regarding basic operations implies Church’s Thesis, namely, that the only numeric functions that can be calculated by effective means are the recursive ones (which are the same, extensionally, as the Turingcomputable numeric functions). In particular, this gives a natural axiomatization of Church’s Thesis, as Gödel and others suggested may be possible.
Arithmetic complexity
 ACM Transactions on Computational Logic
, 2007
"... My purpose in this lecture is to explain how the representation of algorithms by recursive programs can be used in complexity theory, especially in the derivation of lower bounds for worstcase time complexity, which apply to all—or, at least, a very large class of—algorithms. It may be argued that ..."
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My purpose in this lecture is to explain how the representation of algorithms by recursive programs can be used in complexity theory, especially in the derivation of lower bounds for worstcase time complexity, which apply to all—or, at least, a very large class of—algorithms. It may be argued that recursive programs are not a new computational paradigm, since their manifestation as HerbrandGödelKleene systems was present at the very beginning of the modern theory of computability, in 1934. But they have been dissed as tools for complexity analysis, and part of my mission here is to rehabilitate them. I will draw my examples primarily from van den Dries ’ [1] and the joint work in [3, 2], incidentally providing some publicity for the fine results in those papers. Some of these results are stated in Section 3; before that, I will set the stage in Sections 1 and 2, and in the last Section 4 of this abstract I will outline very briefly some conclusions about recursion and complexity which I believe that they support. 1 Partial Algebras